How is the gambler’s fallacy not a logical paradox? A flipped coin coming up heads 25 times in a row has odds in the millions, but if you flip heads 24 times in a row, the 25th flip still has odds of exactly 0.5 heads. Isn’t there something logically weird about that?

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I know it’s true, it’s just something that seems hard to wrap my head around. How is this not a logical paradox?

In: Mathematics

30 Answers

Anonymous 0 Comments

Each turn you have a 50% chance of getting a heads or tails. Getting multiple heads in a row has an extremely low probability. But on any single turn, regardless of the pattern of heads or tails that came before it, the chance of getting either heads or tails remains at 50% each.

Anonymous 0 Comments

Everyone’s phone number is in pi because pi is so big.

Every time you flip a fair coin, it’s a new flip. Eventually there will be a result that you want, but who knows when it will occur?

Anonymous 0 Comments

The easiest way to look at this is on a grand time scale, say 1 trillion outcomes (or infinity if it suits your fancy). During that amount of flips it should not seem unusual that you could have several million heads flipped in a row and the same for tails at some point. Probabilities are always figured so that the outcome gets closer to the theoretical (0.50 in this case) as you approach infinity. Taking a small sample size of 25 means nothing.

Anonymous 0 Comments

Ish?

Which is more random:

HHHHTHTTHTTHHTHTHTT

OR

HHHHHHHHHHHHHHHHH

OR

HTHTHTHTHTHTHTHTHTH

?

They are all equally random. The chance of flipping a million heads in a row, is just as likely to occur as any million state sequence of coin flips. They occur at PRECISELY the same rate. Namely, almost never.

That being said, what’s the chance that if you flip it a million times, you’ll achieve SOME 1 million state sequence?

100%.

SOME sequence WILL result. (Note: SEQUENCE, not PATTERN)

Astronomically improbable things happen regularly. Almost exclusively. There are more probable things that happen, than probable things that happen.. since there are theoretically only a finite number of improbable things, and an infinite number of astronomically improbable things that could happen.

No SPECIFIC improbable thing ever happens. But it would be infinitely impossible for astronomically improbable things to NEVER happen. Given the sheer volume.

Matt Parker did a video on this… but I’m running late to a family function… If I find it, I’ll post it…

Anonymous 0 Comments

No. Think about it like this. Take any coin. I have a Roman coin from CE 156. I’m going to flip it once. What are the odds of it being heads? 50:50 – easy, right? Wouldn’t it be weird if the odds of that one flip depended on how many times the coin had been flipped in the past and whether it came up heads or tails each time?

Anonymous 0 Comments

I think this has been answered pretty well by others, but if any visual learners are having trouble grasping the concept, I present to you this:

Imagine a [Binary Tree](https://en.wikipedia.org/wiki/Binary_tree). Each node is our coin at any given flip, and the two branches from each node represent heads and tails. Flipping our coin at any given node, there is a 50% chance of heads or tails, since there are only two possible outcomes. But ending up at a specific node (following the exact path of heads and tails) has a much smaller chance of happening the more we flip our coin. But remember, no matter how small the odds were of getting to a node, there are still only two equally likely possibilities forward – heads or tails.

So yeah, if we had to bet on landing on a specific node 25 flips later on the tree only knowing the root, our chances of guessing correctly would suck. But if we have to bet on which node we hit next, we only have two options, so our chances are 50/50.

Anonymous 0 Comments

The problem is not with stats, it’s with psychology.

Formation of life was impossible. Chances of a hole in one in golf are near impossible. Chances of winning a lottery are literally zero. But these things actually happened, right – **AND** – often enough!?

So your mind decides: Randomness is NOT random! I can find out some situations where it is NOT random at all!! The sniff of spotting that rare occurrence when random/unordered events are now due for a predictable order for a short while which YOU have spotted.

Our mind decides that it CAN figure out an order even in randomness. That’s what makes you do all kinda dumb shit.

A coin has no memory; but you with the memory of 24 consecutive heads do have a memory – so the coin is no longer random – it’s starting to get in tune – or IS in tune – with your memory of how randomness gotta catch up sooner or later.

But randomness has no memory. It has no idea when to catch up.

Anonymous 0 Comments

The thing about coin flipping is that there are only two possible states it could realistically land on: heads or tails. Even though initially your results may show it favoring one side over the other, the longer you perform the experiment, the closer the average result is going to be 50%.

Anonymous 0 Comments

I have always been god awful at math and things like this.

So my question is this. Lets say the coin is a perfect 50/50 “fair” coin. It was flipped until it hit either heads or tails 24 times in a row. You have to pick heads or tails. I know its 50/50 on the flip.

They offer you the information on if the last 24 were all heads or all tails if you want it. Is there ANY benefit to this?

I am going to say maybe for fun I would choose the opposite. But I know its still 50/50. The sequences in coin flips if you flipped them a billion times or whatever im sure you would have some strange patterns according to humans.

Is there such thing as betting against the sequence continuing, and does that give me better odds in any theoretical way?

Anonymous 0 Comments

I find it easier to comprehend it this way:

So you have an old but otherwise regular coin. You flip it. What are the chances of a heads? 50% right? Now, remember, this coin has been around for a while, exchanged hands, so it’s probably been flipped a good number of times, right? Your flip wasn’t its first flip. It could’ve been flipped thousands of times for all you know. But you have no problem with believing it’s still 50/50 because you simply didn’t consider its past results, right? Well, that’s the thing, statistics never considers the last results either.